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In a forward to a book on Groups of Prime Power Order, Z. Janko says

Of special interest are $2$-groups. In fact, if $G$ is a non-abelian finite simple group and if the structure of its Sylow-$2$ subgroup is know, then the structure of $G$ is almost uniquely determined.

However, the book is written on groups of prime power order, it does not contain any illustrative example of this fact.

Can one give example(s) of infinite family of finite simple groups, which can be almost (in Janko's sense) characterized from structure of Sylow-$2$ subgroups?

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It seems to me that the reference there is to some classical theorems on the matter. See, for example, the Walter theorem or the Alperin–Brauer–Gorenstein theorem.

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  • $\begingroup$ Also, the Gorenstein-Walter Theorem, for example. $\endgroup$ – Geoff Robinson Oct 19 '15 at 16:05

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